INTRODUCTION TO HODGE THEORY

DANIEL MATEI SNSB 2008

Abstract. This course will present the basics of Hodge theory aiming to familiarize students with an important technique in complex and . We start by reviewing complex manifolds, Kahler manifolds and the de Rham theorems. We then introduce Laplacians and establish the connection between harmonic forms and . The main theorems are then detailed: the Hodge decomposition and the Lefschetz decomposition. The Hodge index theorem, Hodge structures and polariza- tions are discussed. The non-compact case is also considered. Finally, time permitted, rudiments of the theory of variations of Hodge structures are given.

Date: February 20, 2008. Key words and phrases. Riemann manifold, , deRham cohomology, harmonic form, Kahler manifold, Hodge decomposition. 1 2 DANIEL MATEI SNSB 2008

1. Introduction The goal of these lectures is to explain the existence of special structures on the coho- mology of Kahler manifolds, namely, the Hodge decomposition and the Lefschetz decom- position, and to discuss their basic properties and consequences. A Kahler manifold is a complex manifold equipped with a Hermitian metric whose imaginary part, which is a 2-form of type (1,1) relative to the complex structure, is closed. This 2-form is called the Kahler form of the Kahler metric. Smooth projective complex manifolds are special cases of compact Kahler manifolds. As (equipped, for example, with the Fubini-Study metric) is a Kahler manifold, the complex submanifolds of projective space equipped with the induced metric are also Kahler. We can indicate precisely which members of the set of Kahler manifolds are complex projective, thanks to Kodaira’s theorem: Theorem 1.1. A compact complex manifold admits a holomorphic embedding into com- plex projective space if and only if it admits a Kahler metric whose Kahler form is of class. We are essentially interested in the class of Kahler manifolds, without particularly emphasising projective manifolds. The reason is that our goal here is to establish the existence of the Hodge decomposition and the Lefschetz decomposition on the cohomology of such a manifold, and for this, there is no need to assume that the Kahler class is integral. However, the Lefschetz decomposition will be defined on the rational cohomology only in the projective case, and this is already an important reason to restrict ourselves, later, to the case of projective manifolds. If X is a complex manifold, the tangent space to X at each point x is equipped with a complex structure Jx. The data consisting of this complex structure at each point is what is known as the underlying almost complex structure. The Jx provide a decomposition 1,0 0,1 (1.1) TxX ⊗ C = TxX ⊕ TxX , 1,0 where TxX is the of complexified tangent vectors u such that Jxu = iu 0,1 1,0 and TxX is the complex conjugate of TxX . From the point of view of the complex structure, i.e. of the local data of holomorphic coordinates, the vector fields of type (0, 1) are those which kill the holomorphic functions. The decomposition (1.1) induces a similar decomposition on the bundles of complex differential forms (1.2) ΛkT ∗X ⊗ C = Λp,qT ∗X, p+Mq=k where Λp,qT ∗X = ΛpT ∗X1,0 ⊗ ΛqT ∗X0,1 and T ∗X ⊗ C = T ∗X1,0 ⊕ T ∗X0,1, is the dual decomposition of (1.1). INTRODUCTIONTOHODGETHEORY 3

The decomposition (1.2) has the property of Hodge symmetry Λp,qT ∗X = Λq,pT ∗X, where complex conjugation acts naturally on ΛkT ∗X ⊗ C. k If we let E (X)C denote the space of complex differential forms of degree on X, i.e. the C∞-sections of the ΛkT ∗X ⊗ C, then we also have the exterior differential k k+1 d : E (X)C →E (X)C, which satisfies d ◦ d = 0. We then define the kth group of X by

k k Ker(d : E (X)C →E +1(X)C) Hk(X, C)= . Im(d : Ek−1(X)C →Ek(X)C) The main theorem proved in these notes is the following. Theorem 1.2. Let Hp,q(X) ⊂ Hk(X, C) be the set of classes which are representable by a closed form α which is of type (p,q) at every point x in the decomposition (1.2). Then we have a decomposition (1.3) Hk(X, C)= Hp,q(X). p q k +M= Note that by definition, we have the Hodge symmetry Hp,q(X)= Hq,p(X) where complex conjugation acts naturally on Hk(X, C)= Hk(X, R) ⊗ C. Here Hk(X, R) is defined by replacing the complex differential forms by real differential forms in the above definition. This theorem immediately gives constraints on the cohomology of a Kahler manifold, which reveal the existence of compact complex manifolds which are not Kahler. For example, the decomposition (1.3) and the Hodge symmetry imply that the dimensions k bk(X) := dimC H (X, C), called the Betti numbers, are even for odd k, property not satisfied by Hopf surfaces. Example 1.3. The Hopf surfaces are the quotients of C2 \ {0} by the fixed-point-free action of a group isomorphic to Z, where a generator g acts via g(z1, z2) = (λ1z1, λ2z2) where the λi, are non-zero complex numbers of modulus strictly less than 1. These surfaces are compact, equipped with the quotient complex structures, and their π1 is isomorphic to Z since C2 \ {0} is simply connected. Thus, their first is equal to 1, which implies that they are not Kahler. The Lefschetz decomposition is another decomposition of the cohomology of a compact Kahler manifold X, this time of topological nature. It depends only on the cohomology class of the Kahler form [ω] ∈ H2(X, R). The exterior product on differential forms satisfies Leibniz’ rule d(α ∧ β)= dα ∧ β + (−1)deg αα ∧ dβ, so the exterior product with ω sends closed forms (i.e. forms killed by d) to closed forms and exact forms (i.e. forms in the image of d) to exact forms. Thus it induces an operator, called the Lefschetz operator, L : Hk(X, R) → Hk+2(X, R). 4 DANIEL MATEI SNSB 2008

The following theorem is sometimes called the hard Lefschetz theorem. Theorem 1.4. For every k ≤ n = dim X, the map (1.4) Ln−k : Hk(X, R) → H2n−k(X, R) is an isomorphism. Remark 1.5. Note that the spaces on the right and on the left are of the same dimension by Poincare duality, which is valid for all compact oriented manifolds. A very simple consequence of the above isomorphism is the following result, which is an additional topological constraint satisfied by Kahler manifolds. Corollary 1.6. The morphism L : Hk(X, R) → Hk+2(X, R) is injective for k ≤ n = dim X. Thus, the odd Betti numbers b2k−1(X) increase with k for 2k − 1 ≤ n, and similarly, the even Betti numbers b2k(X) increase for 2k ≤ n. An algebraic consequence of Lefschetz’ theorem is the Lefschetz decomposition, which as we noted earlier is particularly important in the case of projective manifolds. Let us define the primitive cohomology of a compact Kahler manifold X by k n−k+1 k 2n−k+2 H (X, R)prim := Ker(L : H (X, R) → H (X, R)) for k ≤ n. One can extend this definition to the cohomology of degree >n by using the isomorphism (1.4). Theorem 1.7. The natural map k−2r k i : H (X, R)prim → H (X, R) k−M2r≥0 r (αr) → L αr is an isomorphism for k ≤ n. X Once again, we can extend this decomposition to the cohomology of degree > n by using the isomorphism (1.4). Let us now express the main principle of Hodge theory, which has immense applications. The study of the cohomology of Kahler manifolds and the proof of the Theorems (eq:ihdec) and (1.4), which are the main content of these lectures, are among the most important applications, but the principle applies in various other situations. We restrict ourselves here to giving an explanation of the main idea, which is the notion of a harmonic form, and the application of the theory of elliptic operators which makes it possible to represent the cohomology classes by harmonic forms, but we will omit the proof of the fundamental theorem on elliptic operators, which uses estimations and notions from analysis (Sobolev spaces), which are in different directions from the aims of this book. The delicate point consists in passing from spaces of L2 differential forms, in which the Hodge decomposition is algebraically obvious, to spaces of C∞ differential forms. One of the problems we INTRODUCTIONTOHODGETHEORY 5 encounter is the fact that the operators considered here are differential operators, and thus do not define continuous operators on the spaces of L2 forms. The idea that we want to explain here is the following: using the metric on X, we can define the L2 metric on the spaces of differential forms

(α, β)L2 = hα, βix Vol, ZX where α, β are differential forms of degree k and the scalar product hα, βix at a point x ∈ X is induced by the evaluation of the forms at the point x and by the metric at the point x. The operator d : Ek(X) →Ek+1(X) is a differential operator, and we can construct its formal adjoint d∗ : Ek(X) →Ek−1(X), which is also a differential operator, and satisfies the identity ∗ (α, dβ)L2 = (d α, β)L2 , for α ∈Ek(X), β ∈Ek−1(X). This adjunction relation only makes d∗ into a formal adjoint, since these operators are not defined on the of L2 differential forms, which is the completion of E∗(X) for the L2 metric. The idea of Hodge theory consists in using the adjoint d∗ to write the decompositions Ek(X)=Im d ⊕ Im d⊥ = Im d ⊕ Ker d∗, Ek(X) = Ker d ⊕ Ker d⊥ = Ker d∗ ⊕ Im d, and finally, using the inclusion Im d ⊂ Ker d, Ek(X)=Im d ⊕ Im d∗ ⊕ Ker d ∩ Ker d∗. Of course, these identities, which would be valid on finite-dimensional spaces or Hilbert spaces since the operator d has closed image there, require the analysis mentioned above in order to justify them here. Apart from this issue, if we accept these identities, we see that the space Hk := Ker d ∩ Ker d∗ ⊂Ek(X) of harmonic forms projects bijectively onto Hk(X, R) (or Hk(X, C) if we study the co- homology with complex coefficients), since it is a supplementary space of Im d inside Ker d. Another characterisation of harmonic forms uses the Laplacian ∗ ∗ ∆d = dd + d d. Indeed, it is very easy to see that we have k H = Ker∆d.

The operator ∆d is an . This property of a differential operator can be read directly from its symbol, which is essentially its homogeneous term of largest order (which is 2 for the Laplacian). The decompositions written above are special cases of the decomposition associated to an elliptic operator. The Hodge decomposition (1.3) is obtained by combining the Hodge theory sketched above and the study of the properties of the Laplacian of a Kahler manifold. We have already mentioned various operators acting on the spaces of differential forms of a Kahler 6 DANIEL MATEI SNSB 2008 manifold, namely d, L and their formal adjoints d∗, Λ for the L2 metric. Moreover, the complex structure makes it possible to decompose d as d = ∂ + ∂,¯ where the Dolbeault operator ∂¯ sends α ∈Ep,q(X) to the component of bidegree (p,q + 1) of dα. Here α ∈ Ep,q(X) is the space of differential forms of bidegree (p,q) at every point of X; it is also the space of sections of the bundle Λp,qT ∗X, which appears in the decomposition (1.2) given by the complex structure. The differential operators ∂ and ∂¯ are differential operators of order 1, and have formal adjoint operators ∂∗ and ∂¯∗. The Kahler identities establish commutation relations between these operators. For example, we have the identity [Λ, ∂]= i∂¯∗, and the other identities follow from this one via passage to the complex conjugate or to the adjoint. From these identities, and from the fact that L commutes with d while ∂ and ∂¯ anticommute, we deduce the following result. ¯ Theorem 1.8. The Laplacians ∆d, ∆∂ and ∆∂¯ associated to the operators d, ∂ and ∂ respectively satisfy the equalities

(1.5) ∆d = 2∆∂ = 2∆∂¯. We deduce that the harmonic forms for d are also harmonic for ∂ and ∂¯, and in particular are also ∂ and ∂¯-closed. Finally, as the operators ∂ and ∂¯ are bihomogeneous (of bidegree (1, 0) and (0, 1) respectively) for the bigraduation of the spaces of differential forms given by the decomposition (1.2), it follows easily that each of the Laplacians ∆∂ and ∆∂¯ is bihomogeneous of bidegree (0, 0), i.e. preserves the forms of type (p,q) for every bidegree (p,q). The same then holds for ∆d by the equality (1.5). The Hodge decomposition is then obtained simply by the decomposition of the harmonic forms as sums of forms of type (p,q): Corollary 1.9. Let X be a compact Kahler manifold. If α is a harmonic form (for the Laplacian associated to the operator d and to the metric), its components of type (p,q) are harmonic. Thus, we have a decomposition (1.6) Hk(X)= Hp,q(X), p+Mq=k where Hp,q(X) is the space of harmonic forms of type (p,q) at every point of X. The Hodge decomposition (1.3) is obtained by combining the theorem of representation of cohomology classes by harmonic forms with the decomposition (1.6). The Lefschetz decomposition is also an easy consequence of the decomposition (1.6). Indeed, we first show that Theorem 1.4 holds for the operator L acting on differential forms. Furthermore, the Kahler identities show that L commutes with the Laplacian, so that the operators Lr send harmonic forms to harmonic forms, and once the theorem is proved on the level of forms, it remains valid on the level of harmonic forms, and thus also on cohomology classes. INTRODUCTIONTOHODGETHEORY 7

The Hodge decomposition (1.3) gives an extremely interesting structure when it is k k combined with the integral structure on the cohomology H (X, C) = H (X, Z) ⊗Z C. For this equality, which follows from the change of coefficients theorem, one must adopt a different definition of cohomology, which does not make use of differential forms. For k Z one possible definition, we can introduce the singular cohomology Hsing(X, ). We start from the complex (C∗(X), ∂ : Ck(X) → Ck−1(X)) of singular chains, where Ck(X) is the free abelian group generated by the continuous maps from the simplex ∆k of dimension k to X. The map ∂ is given by the restriction to the boundary i ∂φ = (−1) φ|∆k,i i X th ∗ where ∆k,i is the i face of ∆k. The complex (Csing(X), d) of singular cochains is then defined as the dual complex of (C∗(X), ∂). Its cohomology is the singular cohomology ∗ Z Hsing(X, ). We have the following theorem, due to de Rham. K R C k K k Z K Theorem 1.10. For = or , we have H (X, )= Hsing(X, ) ⊗Z . If we consider the complex of differentiable chains, we can prove this theorem by using k ∗ the natural map from E (X) to Csing(X) given by integration:

α → φ → φ∗α .  Z∆k  A much more conceptual proof of de Rham’s theorem can be given by using the lan- guage of theory. will be used, for example, in the Hodge decom- p,q q p position, to describe the spaces H as the groups H (X, ΩX ), which are defined for every complex manifold X as the qth cohomology group of X with p values in the sheaf ΩX of holomorphic differential forms of degree p. We note, however, that this identification is valid only in the Kahler case. In general, without the Kahler q p hypothesis, we cannot identify H (X, ΩX ) with the space of cohomology classes of degree p + q which are representable by a closed form of type (p,q) at every point. Let Γ be the functor of global sections Γ of the category of sheaves of abelian groups on X to the category of abelian groups. We show using Poincare’s theorem that the sheaves of differential forms form a Γ-acyclic resolution of the constant sheaf CX (often written C) of stalk C, so that the space Hk(X, C) defined above must be understood as th the k cohomology group of X with values in CX . Similarly, we can interpret the singular cohomology as the cohomology of the complex of global sections of a Γ-acyclic resolution Z ∗ Z k Z of the constant sheaf of stalk . Thus, we have Hsing(X, ) = H (X, ) canonically. De Rham’s theorem thus reduces to proving a change of coefficients theorem for the k k cohomology of the sheaves H (X, C)= H (X, Z) ⊗Z C, which is not difficult. These different interpretations of the cohomology, corresponding to different resolu- tions, are all equally important, since they carry different types of information. For example, the Hodge decomposition of the cohomology of a Kahler manifold requires the de Rham version of the cohomology, while that of the integral structure requires another version, singular or Cech for example. 8 DANIEL MATEI SNSB 2008

2. Holomorphic Functions of Many Variables In this section, we recall the main properties of holomorphic functions of several com- plex variables. The C-valued holomorphic functions of the complex variables z1,...,zn those whose differential is C-linear, or equivalently, those which are annihilated by the operators ∂ . It follows immediately from this definition that the set of holomorphic ∂z¯l functions forms a ring, and that the composition of two holomorphic functions is holo- morphic. The following theorem, however, requires a certain amount of work.

Theorem 2.1. The holomorphic functions of the complex variables z1,...,zn are complex analytic, i.e. they locally admit expansions as power series in the variables zl. This result is an easy consequence of Cauchy’s formula in several variables, which is a generalisation of the formula 1 f(ζ) f(z)= dζ, 2πi ζ − z Z|ζ|=1 where f is a defined in a disk of radius > 1, and |z| < 1. Cauchy’s formula can also be used to prove Riemann’s theorem of analytic continuation: Theorem 2.2. Let f be a bounded holomorphic function on the pointed disk. Then f extends to a holomorphic function on the whole disk. And also Hartogs’ theorem: Theorem 2.3. Let f be a holomorphic function defined on the complement of an analytic subset of codimension 2 in a ball B of Cn,n ≥ 2. Then f extends to a holomorphic function on B. Hartogs’ theorem enables us to show that a holomorphic section of a complex vector bundle over a complex manifold is defined everywhere if it is defined on the complement of an analytic subset of codimension 2. 2.4. Holomorphic functions of one variable. Let U ⊂ C = R2 be an open set, and f : U → C a C1 map. Let x,y be the linear coordinates on R2 such that z = x + iy is the canonical linear complex coordinate on C. Consider the complex-valued differential form

dz = dx + idy ∈ HomR(T U, C)=ΩU,R ⊗ C.

Clearly dz and its complex conjugate dz¯ form a basis of ΩU,R ⊗ C over C at every point of U, since 2dx = dz + dz,¯ 2idy = dz − dz.¯ The complex differential form df ∈ HomR(T U, C) can thus be uniquely written ∂f ∂f d f = (u)dz + (u)dz,¯ u ∂z ∂z¯ ∂f ∂f where the complex-valued functions ∂z (·), ∂z¯ (·) are continuous. We obviously have ∂f 1 ∂f ∂f ∂f 1 ∂f ∂f = − i , = + i . ∂z 2 ∂x ∂y ∂z¯ 2 ∂x ∂y     INTRODUCTIONTOHODGETHEORY 9

We can also consider the above decomposition of df as the decomposition of df ∈ Hom(C, C) into C-linear and C-antilinear parts: ∂f R C C Lemma 2.5. We have ∂z¯ (u) = 0 if and only if the -linear map duf : TuU = → is C-linear, i.e. is equal to multiplication by a complex number, which is then equal to ∂f ∂z (u). Definition 2.6. The function f is said to be holomorphic if it satisfies one of the equiv- alent conditions of lemma 2.5 at every point of U. Lemma 2.7. If f is holomorphic and does not vanish on U, then 1/f is holomorphic. Similarly, if f, g are holomorphic, fg and f + g and g ◦ f (when g is defined on the image of f) are all holomorphic. In particular, we will use the following corollary. Corollary 2.8. If f is holomorphic on U, the map g defined by f(z) g(z)= z − a is holomorphic on U \ {a}. 2.9. Stokes’formula. Let α be a differential k-form on an n-dimensional manifold U. If x1,...,xn are local coordinates on U, we can write

α = αI dxI , I X where the indices I parametrise the totally ordered subsets i1 < · · · < ik of {1, ..., n}, with dxI = dxi1 ∧···∧ dxik . We can then define the continuous (k + 1)-form

∂αI (2.1) dα = dxi ∧ dxI ; ∂xi I,i X we check that it is independent of the choice of coordinates. This follows from the more general fact that if V is an m-dimensional manifold and φ : V → U is a differential map ∗ given in local coordinates by φ xi := xi ◦ φ = φ(y1,...,ym), then for every differential

form α = I αI dxi1 ∧···∧ dxik , we can define its inverse image ∗ P φ α = αI ◦ φ dφi1 ∧···∧ dφik . I X Moreover, this image inverse satisfies d(φ∗α)= φ∗(dα),

where the coordinates yi (and the formulae (2.1)) are used on the left, while the coordi- nates xi, are used on the right. A differential k-form a can be integrated over the compact oriented k-dimensional submanifolds of U with boundary, or over the image of such manifolds under differentiable maps. 10 DANIEL MATEI SNSB 2008

To begin with, let us recall that a k-dimensional manifold with boundary is a covered by open sets Ui which are homeomorphic, via certain maps hi to open subsets of Rk or to (0, 1] × V , where V is an open set of Rk−1. We require the transition −1 functions hi ◦ hj to be differentiable on hj(Ui ∩ Uj). When hj(Ui ∩ Uj) contains points on the boundary of Uj, i.e. is locally isomorphic to (0, 1] × V , where V is an open set k−1 of R , hj(Ui ∩ Uj) must also be locally isomorphic to (0, 1] × W , where W is an open Rk−1 −1 set of , and the differentiable map hi ◦ hj must locally extend to a diffeomorphism of a neighbourhood in Rk of (0, 1] × V to a neighbourhood of (0, 1] × W , inducing a diffeomorphism from 1 × V to 1 × W . In particular, the boundary of M, which we denote −1 by ∂M and which is defined, with the preceding notation, as the union of the hi (1 × V ), is a closed set of M which possesses an induced differentiable manifold structure. The manifold with boundary M is said to be oriented if the diffeomorphisms hi ◦ −1 hj have positive Jacobian. The boundary of M is then also naturally oriented by the charts 1 × V , where V is an open set of Rk−1 as above, since the induced transition −1 diffeomorphisms hi ◦ hj |1×V : V → W also have positive Jacobian. If M is k-dimensional manifold with boundary and φ : M → U is a differentiable map (along the boundary of M, which is locally isomorphic to (0, 1] × V , we require φ to extend locally to a differentiable map on a neighbourhood (0, 1+ ǫ) × V of {1}× V ), then for every differentiable k-form α, we have the inverse image β = φ∗α defined above, which is a differentiable k-form on M. If moreover M is oriented and compact, such a form can be integrated over M as follows. Let ρi be a partition of unity subordinate to a covering of M by open sets Ui as above, which we may assume to be diffeomorphic to k−1 k (0, 1] × (0, 1) or to (0, 1) . Then β = i ρiβ on extends to a differentiable form on k [0, 1] . Setting ρiβ = gi(x ,...,x )dx ∧···∧ dx , we then define 1 k 1 P k 1 1 β = ρiβ, ρiβ = . . . gi(x1,...,xk)dx1 ...dxk. M i Ui Ui 0 0 Z X Z Z Z Z The change of variables formula for multiple and the fact that the authorised variable changes have positive Jacobians ensure that M β is well-defined independently of the choice of oriented charts, i.e. of local orientation-preserving coordinates. R Remark 2.10. If we change the orientation of M, i.e. if we compose all the charts with Rk ∗ local diffeomorphisms of with negative Jacobians, the integrals M φ α change sign. This follows from the change of variables formula for multiple integrals, which uses only the absolute value of the Jacobian, whereas the change of variables formulaR for differential forms of maximal degree uses the Jacobian itself.

Suppose now that α is a (k − 1)-form on U. Then, as φ |∂M is differentiable and ∂M is ∗ a compact oriented manifold of dimension k − 1, we can compute the integral ∂M φ α. Moreover, we can integrate the differential dφ∗α = φ∗dα over M. We then have R Theorem 2.11. (Stokes’ formula) The following equality holds:

φ∗dα = φ∗α. ZM Z∂M INTRODUCTIONTOHODGETHEORY 11

∗ In particular, if dα = 0, we have ∂M φ α = 0. We will use Stokes’ formula veryR frequently throughout this text. In particular, it will enable us to pair the de Rham cohomology with the . The following consequence will be particularly useful. Corollary 2.12. If α is a differential form of degree n − 1 on a compact n- dimensional manifold without boundary, then M dα = 0. 2.13. Cauchy’s formula. We proposeR to apply Stokes’ formula, using the following lemma. Lemma 2.14. Let f : U → C be a holomorphic map. Then the complex differential form fdz is closed. We thus also have the following. Corollary 2.15. If f is holomorphic on U, the differential form f dz is closed on z−z0 U \ {z0}.

Suppose now that U contains a closed disk D. For every z0 ∈ D, let Dǫ be the open disk of radius ǫ centred at 0 which is contained in D for sufficiently small ǫ. Then D\Dǫ is a manifold with boundary, whose boundary is the union of the circle ∂D and the circle of centre z0 and radius ǫ, the first with its natural orientation, the second with the opposite orientation. For holomorphic f, Stokes’ formula and previous corollary then give the equality 1 f(z) 1 f(z) dz = dz. 2πi z − z 2πi z − z Z∂D 0 Z|z−z0|=ǫ 0 Furthermore, we have the following.

Lemma 2.16. If f is a function which is continuous at z0, then 1 f(z) lim dz = f(z0). ǫ→0 2πi z − z Z|z−z0|=ǫ 0 Combining thus lemma and the above equality, we now have Theorem 2.17. (Cauchy’s formula) Let f be a holomorphic function on U and D a closed disk contained in U. Then for every point z0 in the interior of D, we have the equality 1 f(z) f(z )= dz. 0 2πi z − z Z∂D 0 2.18. Holomorphic functions of several variables. 2.18.1. Cauchy’s formula and analyticity. Let U be an open set of Cn, and let f : n U → C be a differentiable map. For u ∈ U, we have a canonical identification TuU = C . We can thus generalise the notion of a holomorphic function to higher dimensions. Definition 2.19. The function f is said to be holomorphic if for every u ∈ U, the n differential duf : Hom(TuU, C) = Hom(C , C) is C-linear. 12 DANIEL MATEI SNSB 2008

It is easy to prove that lemma 2.7 remains true in higher dimensions. Furthermore, we have the three following characterisations of holomorphic functions. Theorem 2.20. The following three properties are equivalent for a differentiable function (i) f is holomorphic. (ii) In the neighbourhood of each point z0 ∈ U, f admits an expansion as a power I series of the form f(z0 + z) = I aI z , where I runs through the set of the n- tuples of integers (i ,...,i ) with i ≥ 0, and zI := zi1 ...zin . The coefficients of 1 n P k 1 n the series satisfy the following property: there exist R1 > 0,...,Rn > 0 such that I the power series I aI r converges for every r1 < R1,...,rn < Rn. (iii) If D = {(ζ1,...,ζn)||ζi − ai| ≤ ǫi} is a poly disk contained in U, then for every P ◦ z = (z1,...,zn) ∈ D , we have the equality 1 n dζ dζ f(z)= f(ζ) 1 ∧···∧ n . 2πi ζ − z ζ − z   Z|ζi−ai|=ǫi 1 1 n n In the preceding formula, the integral is taken over a product of circles, equipped with the orientation which is the product of the natural orientations. Because of property (ii), holomorphic functions are also known as complex analytic functions. Property (iii) is Cauchy’s formula in several variables. One can prove it by induction on the dimension, using Cauchy’s formula in a single variable. One can also directly apply Stokes’ formula, using the following analogue of lemma 2.14.

Lemma 2.21. If f is holomorphic, then the differential form f(z)dz1 ∧···∧dzn is closed. Let us give some applications of theorem 2.20. To begin with, we have Theorem 2.22. (The maximum principle) Let f be a holomorphic function on an open subset U of Cn. If |f| admits a local maximum at a point u ∈ U, then f is constant in the neighbourhood of this point. Another essential application is the principle of analytic continuation. Theorem 2.23. Let U be a connected open set of Cn, and f a holomorphic function on U. If f vanishes on an open set of U, then f is identically zero. Let us now give some subtler applications of Cauchy’s formula or its generalisations. These theorems show that the possible singularities of a holomorphic function cannot exist unless the function is not bounded (Riemann), and is not defined on the complement of an analytic subset of codimension 2 (Hartogs).

Theorem 2.24. (Riemann) Let f be a holomorphic function on U \ {z|z1 = 0}, where U is an open set of Cn. Then if f is locally bounded on U, f extends to a holomorphic map on U. To conclude this section, we will mention the following version of Hartogs’ extension theorem which implies the more general theorem mentioned above. Theorem 2.25. Let U be an open set of Cn and f a holomorphic function defined on U \ {z|z1 = z2 = 0}. Then f extends to a holomorphic function on U. INTRODUCTIONTOHODGETHEORY 13

∂g 2.26. The equation ∂z¯ = f. The following theorem will play an essential role in the proof of the local exactness of the operator ∂¯. Theorem 2.27. Let f be a differentiable function on an open set of C. Then, locally on this open set, there exists a differentiable function g (defined up to the addition of a holomorphic function), such that ∂g = f. ∂z¯ Proof. As the statement is local, we may assume that f has compact support, and thus is defined on C. Now set 1 f(ζ) g(z)= dζ ∧ dζ.¯ 2πi C ζ − z Z This is a singular integral. By definition, it is equal to the limit, as ǫ → 0, of the integrals 1 f(ζ) g(z)= dζ ∧ dζ,¯ 2πi C ζ − z Z \Dǫ where Dǫ is a disk of radius ǫ centred at z. It is easy to see that this limit exists (the f(ζ) 1 ′ function ζ−z is L ). Making the change of variable ζ = ζ − z, we also have ′ 1 f(ζ + z) ′ ′ g(z) = lim gǫ(z), gǫ(z)= dζ ∧ dζ¯ , ǫ→0 2πi C ′ ζ′ Z \Dǫ The convergence of the gǫ(z) when ǫ → 0 is uniform in z. Moreover, we can differentiate under the integral sign the (non-singular) integral defining gǫ ∂g 1 ∂f(ζ′ + z) dζ′ ∧ dζ¯′ ǫ = , ∂z¯ 2πi C ′ ∂z¯ ζ′ Z \Dǫ ′ ∂f(ζ +z) ∂gǫ As ∂z¯ is differentiable, the functions ∂z¯ converge uniformly, and we conclude that g is differentiable and satisfies ∂g 1 ∂f(ζ′ + z) dζ′ ∧ dζ¯′ = , ∂z¯ 2πi C ∂z¯ ζ′ Z ∂g ′ Thus, it remains to show the equality ∂z¯ = f. Changing back to ζ = ζ + z, we have ∂g 1 ∂f(ζ) dζ ∧ dζ¯ (2.2) = lim , ∂z¯ ǫ→0 2πi C ∂ζ¯ ζ − z Z \Dǫ C ∂f(ζ) dζ∧dζ¯ fdζ Now, on \ Dǫ we have the equality ∂ζ¯ ζ−z = −d ζ−z . Stokes’ formula thus gives 1 ∂f(ζ) dζ ∧ dζ¯ 1   dζ (2.3) = f(ζ) . 2πi C ∂ζ¯ ζ − z 2πi ζ − z Z \Dǫ Z∂Dǫ Using lemma 2.16 and the equalities (2.2), (2.3) we have thus proved the desired equality.  14 DANIEL MATEI SNSB 2008

3. Complex Manifolds We introduce and study the notion of a complex structure on a differentiable or com- plex manifold. A complex manifold X of (complex) dimension n is a differentiable man- ifold locally equipped with complex-valued coordinates (called holomorphic coordinates) n n z1,...,zn, such that the diffeomorphisms from an open set of C to an open set of C given by coordinate changes are holomorphic. By the definition of a holomorphic transforma- tion, we then see that the structure of a complex vector space on the tangent space TxX n given by the identification TxX = C induced by the holomorphic coordinates z1,...,zn does not depend on the choice of holomorphic coordinates. The tangent bundle T X of a complex manifold X is thus equipped with the structure of a complex vector bundle. Such a structure is called an almost complex structure. The Newlander-Nirenberg theo- rem characterises the almost complex structures induced as above by a complex structure. We also introduce holomorphic vector bundles over a complex manifold. These vector bundles are those whose ”transition matrices” are holomorphic. It turns out that we can define a differential operator (the Dolbeault operator) ∂¯ on the space of sections of such a vector bundle E, and more generally, on the space of differential forms with values in such a bundle. The holomorphic sections σ of E are then characterised by the equation ∂σ¯ = 0. One can show that the Dolbeault operator satisfies the condition ∂¯ ◦ ∂¯ = 0, and that the complex defined in this way is locally exact. This will be used later to represent the cohomology of X with values in the sheaf of holomorphic sections of E using ∂¯-closed differential forms with coefficients in E.

3.1. Manifolds and vector bundles.

3.1.1. Definitions. A topological manifold is a topological space X equipped with a covering by open sets Ui, which are homeomorphic, via maps hi called ”local charts”, to open sets of Rn. One can show that such an n is necessarily independent of i when X is connected; n is then called the dimension of X. Definition 3.2. A C∞ differentiable manifold X is a topological manifold equipped with n a system of local charts hi : Ui → R such that the open sets Ui cover X, and the change −1 ∞ of chart morphisms hj ◦ hi : hi(Ui ∩ Uj) → hj(Ui ∩ Uj) are C maps. A C∞ differentiable function on such a manifold (or on an open set) is a function f such −1 ∞ that for each Ui, the function f ◦ hi is C differentiable. A map f : X → Y between ∞ −1 ∞ differentiable manifolds is C differentiable map if gj ◦ f ◦ hi are C differentiable. A real (resp. complex) topological vector bundle of rank m over a topological space X is a topological space E equipped with a map π : E → X such that for an open cover {Ui} of X, we have ”local trivialisation” homeomorphisms

−1 m m φi : π (Ui) → Ui × R (resp. Ui × C ), such that:

(i) pr1 ◦ φi = π INTRODUCTIONTOHODGETHEORY 15

−1 −1 −1 R (ii) The transition functions φj ◦φi : φ(π (Ui ∩Uj)) → φ(π (Ui∩Uj)) are -linear (resp. C-linear) on each fibre x × Rm (resp. x × Cm). m m Such a transformation Ui ∩ Uj × R → Ui ∩ Uj × R must respect the first projection, by condition (i) above, and is thus described by a real m × m matrix, whose coefficients, by continuity, are continuous functions of x ∈ Ui ∩ Uj. (In the complex case, we must consider complex matrices.) These matrices are called transition matrices. Definition 3.3. If X is a differentiable manifold, a vector bundle E over X is equipped with a differentiable structure if we are given local trivialisations whose transition matrices are C∞. Remark 3.4. The bundle E is then equipped with the structure of a C∞ manifold for which π is C∞ as well as the local trivialisations.

A section of a vector bundle π : E → X is a map σ : X → E such that π◦σ = IdX . This section is said to be continuous, resp. differentiable, if a is continuous, resp. differentiable. −1 If π : E → X is a vector bundle and x ∈ X, we write Ex := π (x). It is canonically a vector space, with structure given by any of the trivialisations of E in the neighbourhood of x; Ex is called the fibre of E at the point x. A vector bundle π : E → X is said to be trivial if it admits a global trivialisation φ : E = X × Rn. Equivalently, E must admit n global sections which provide a basis −1 of the fibre Ex at each point. These sections are given by at σi = φ ◦ e˜i, where n n e˜i : X → X × R is given by ei(x) = (x, ei), where the ei form the standard basis of R . Let πE : E → X and πF : F → X be vector bundles over X. A morphism ψ : E → F of vector bundles is a continuous map such that πF ◦ ψ = πE, and ψ is linear on each fibre. This means that in local trivialisations, ψ becomes linear (C-linear in the case of complex bundles) on the fibres x ∈ Rn; his definition is independent of the choice of the open set containing x, since the transition functions are also linear on the fibres. We have an analogous definition for differentiable bundles. Given a vector bundle E, we can define its dual E∗ and its exterior powers ΛkE, which are differentiable if E is. The points of E∗ are the linear forms on the fibres of ∗ πE : E → X; E admits a natural trivialisation when E is trivialised, and the transition matrices of E∗ are the inverses of the transposes of the transition matrices of E. Similarly, the points of ΛkE can be identified with the alternating K-linear forms on the fibres of ∗ ∗ πE : E → X. 3.4.1. The tangent bundle. If X is a differentiable manifold, the tangent bundle T X of X is a differentiable bundle of rank n = dim X which we can define as follows. If X n is covered by open sets Ui equipped with diffeomorphisms hi to open sets of R , then n T X is covered by open sets Ui × R , where the identifications (or transition morphisms) n n n n between Ui ∩ Uj × R ⊂ Ui × R and Ui ∩ Uj × R ⊂ Uj × R are given by

(x, v) → (x, φij∗(x)v). −1 Here hij = hi ◦ hj is the transition diffeomorphism between the open sets hi(Ui ∩ Uj) n and hj(Ui ∩ Uj) of R , and hij∗(x) is its Jacobian matrix at the point x. A section of the tangent bundle of a differentiable manifold is called a vector field. 16 DANIEL MATEI SNSB 2008

There exist two intrinsic ways of describing the elements of the tangent bundle. The points of the tangent bundle can be identified with equivalence classes of differentiable maps γ : [−ǫ,ǫ] → X (for an ǫ ∈ R,ǫ> 0 varying with γ) for the equivalence relation dγ dγ′ γ ≡ γ′ if and only if γ(0) = γ′(0), (0) = (0). dt dt The second equality in this definition makes sense in any local chart for X in the neighbourhood of γ(0). We call these equivalence classes ”jets of order 1”. To check that the set defined in this way has the structure of a vector bundle introduced earlier, it suffices to note that the jets of order 1 of an open set U of Rn can be identified, via the map γ → (γ(0), γ˙ (0)), with U × Rn, and that a diffeomorphism ψ : U → V between two n open sets of R induces the isomorphism (ψ, ψ∗) between the spaces of jets of order 1 of U and V . Another definition of the tangent vectors, i.e. of the elements of the tangent bundle, consists in identifying them with the derivations of the algebra of the real differentiable functions on X with values in R supported at a point x ∈ X. This means that we consider the linear maps δ : C∞(X) → R satisfying Leibniz’ rule δ(fg)= f(x)δ(g)+ g(x)δ(f) for a point x ∈ X. The equivalence between the two definitions is realised by the map d(γ◦f) which to a jet γ associates the derivation δγ (f)= dt (0). Definition 3.5. A differential form of degree k is a section of ΛkT ∗X. We write deg α for the degree of such a form α.

In general, we write ΩX,R the bundle of real differential 1-forms, and ΩX,C for its complexification Hom(ΩX,R, C). Similarly, the bundle of real (resp. complex) k-forms k k is written ΩX,R (resp. ΩX,C). We see immediately that if f is a real differentiable ∞ function on X, then df is a C section of ΩX,R. We also see that if x1,...,xn are

local coordinates defined on an open set U ⊂ X, then the dxI = dxil ∧···∧ dxik , k 1 ≤ i1 < · · · < ik ≤ n provide a basis of the fibre of ΩX,R at each point of the open set U. Indeed, by the definition of T X, the coordinates xi provide a local trivialisation of T X, where the corresponding local basis is given at each point x ∈ U by the derivations ∂ (x). The dx simply form the dual basis of Ω R at each point of U. ∂xi i X, 3.5.1. Complex manifolds. Let X be a differentiable manifold of dimension 2n. Definition 3.6. We say that X is equipped with a complex structure if X is covered by n open sets Ui which are diffeomorphic, via maps called hi, to open sets of C , in such a way −1 that the transition diffeomorphisms hj ◦ hi : hi(Ui ∩ Uj) → hj(Ui ∩ Uj) are holomorphic. The (complex) dimension of X is by definition equal to n. On a complex manifold, a C −1 map f : U → defined on an open set U is said to be holomorphic if f ◦hi is holomorphic on hi(U ∩ Ui). Once again, this definition does not depend on the choice of chart, since the change of chart morphisms is holomorphic and compositions of holomorphic functions are also holomorphic. We can also define the notion of a holomorphic vector bundle. INTRODUCTIONTOHODGETHEORY 17

Definition 3.7. A differentiable complex vector bundle π : E → X over a complex manifold X is said to be equipped with a holomorphic structure if we have trivialisations −1 m φi : π (Ui) → Ui × C , −1 such that the transition matrices φij = φj ◦ φi have holomorphic coefficients. The above trivialisations will be called ”holomorphic trivialisations”. If E is a holo- morphic vector bundle, E is in particular a complex manifold such that π is holomorphic. Indeed, we can assume, in the definition above, that the Ui are charts, i.e. identified n via hi with open sets of C ; then the (hi × IdCn ) ◦ φi give charts for E whose transition functions are clearly holomorphic. A holomorphic section of a holomorphic vector bundle π : E → X over an open set U of X is a section σ : X → E of π which is a holomorphic map. For example, a holomorphic local trivialisation φi of E as above is given by the choice of a family of holomorphic sections of , whose values at each point x of Ui form a basis of the fibre Ex over C. Example 3.8. The holomorphic tangent bundle. This bundle is defined exactly like the real tangent bundle of a differentiable manifold. Given a system of charts hi : Ui → n n n Vi ⊂ C , we define T X as the union of the Ui × C , glued by identifying Ui ∩ Uj × C ⊂ n n n Ui × C and Ui ∩ Uj × C ⊂ Uj × C via

(x, v) → (x, hij∗(x)v).

Here, the holomorphic Jacobian matrix hij∗ is the matrix with holomorphic coefficients ∂hk ij , where h = h ◦ h−1. We can also, as for the real tangent bundle, define the ∂zl ij i j holomorphic tangent bundle as the set of complex-valued derivations of the C-algebra of holomorphic functions, or as the set of jets of order 1 of holomorphic maps from the complex disk to X. 3.9. Integrability of almost complex structures. 3.9.1. Tangent bundle of a complex manifold. Let X be a complex manifold, and let n hk : Ui → C be holomorphic local charts. Then the real tangent bundle T UkR can n be identified, via the differential hk∗ , with Uk × C . Moreover, the change of chart −1 C morphisms hk ◦ hj are holomorphic by hypothesis, i.e. have -linear differentials, for n n n the natural identifications: TxC = C , x ∈ C . It follows that the R-linear operators n Jk : T UkR → T UkR identified with 1 × i acting on Uk × C , glue together on Uk ∩ Uj and define a global endomorphism, written J, of the bundle T XR. Obviously J satisfies the identity J 2 = −1; thus J defines the structure of a C-vector space of rank n on each fibre TxXR. The differentiability of J even shows that T XR is thus equipped with the structure of a differentiable complex vector bundle. This leads us to introduce the following definition. Definition 3.10. An almost complex structure on a differentiable manifold X is an en- domorphism J of the real tangent bundle T XR such that J 2 = −1; equivalently, it is the structure of a complex vector bundle on T XR. We saw that a complex structure on X naturally induces an almost complex structure. 18 DANIEL MATEI SNSB 2008

Definition 3.11. An almost complex structure J on a manifold X is said to be integrable if there exists a complex structure on X which induces J.

In the case of a complex manifold, the relation between T XR seen as a complex vector bundle and the holomorphic tangent bundle T X of X is as follows: the bundle T X is generated, in a chart U, by the elements ∂ 1 ∂ ∂ = − i , ∂z 2 ∂x ∂y j  j j  which are naturally elements of T UR ⊗ C. Thus, in fact, we have an inclusion of complex vector bundles T X ⊂ T XR ⊗ C. Moreover, for an almost complex manifold (X, J), the complexified tangent bundle , T XR ⊗ C contains a complex vector subbundle, denoted by T 1 0X and defined as the bundle of eigenvectors of J for the eigenvalue i. As a real vector bundle, T 1,0X is naturally isomorphic to T XR via the application Re (real part) which to a complex field v + iw associates its real part v. Moreover, this identifies the operators i on T 1,0X and J on , T XR. Clearly T 1 0X is generated by the v − iJv, v ∈ T XR. n n n 2n Furthermore, in the case where X = C , consider the isomorphism T CR = C × R given by the sections ∂ , ∂ of the tangent bundle of Cn, where z = x + iy and z are ∂xj ∂yj k k k k n n complex linear coordinates on C . The induced complex structure operator J on T CR sends ∂ to ∂ . Thus, the tangent vectors of type (1, 0) are generated over C at each ∂xj ∂yj point by the ∂ ∂ ∂ − iJ = 2 . ∂x ∂y ∂z  j j  j In conclusion, we have shown the following. Proposition 3.12. If X is a complex manifold, then X admits an almost complex struc- , ture, and the subbundle T 1 0X ⊂ T XR ⊗ C defined by J is equal, as a complex vector subbundle of T XR ⊗ C, to the holomorphic tangent bundle T X.

Complex conjugation acts naturally on the complexified tangent bundle T XC = T XR ⊗ C of a differentiable manifold X. If J is an almost complex structure on X, we have the , , subbundle T 0 1X of T XC, defined as the complex conjugate of T 1 0X. We can also define it as the set of the complexified tangent vectors which are the eigenvectors of J associated to the eigenvalue −i. Thus, it is clear that we have a direct sum decomposition

1,0 0,1 T XC = T X ⊕ T X. Remark 3.13. When X is an almost complex manifold, the vector bundle T 1,0X does not a priori have the structure of a holomorphic bundle. In what follows, if X is a complex manifold, a section of T X will be taken to mean a holomorphic section of T X, while a section of T 1,0X will be a differentiable section. INTRODUCTIONTOHODGETHEORY 19

If ψ : X → Y is a holomorphic map between two complex manifolds, we define a morphism of holomorphic vector bundles ψ∗ : T X → TY in the obvious way. In holo- morphic local charts which trivialise T X and TY , the matrix of ψ∗ is given by the holomorphic Jacobian matrix ∂ψk of ψ. This morphism can in fact be identified with the ∂zj morphism of real vector bundles ψ∗,R : T XR → TYR via the identifications of real bundles T X = T XR,TY = TYR given by the real part Re. As a morphism of complex bundles, ψ∗ can be deduced from ψ∗,R by noting that ψ∗,R is compatible with the almost complex structures of X and Y , since ψ is holomorphic, and thus induces a C-linear morphism 1,0 1,0 1,0 ψ∗ : T X → T Y . 3.13.1. The Newlander-Nirenberg theorem. Note first that the bracket of vector fields over a differentiable manifold X extends by C-linearity to the complexified vector fields, i.e. to the differentiable sections of T XC. Now, let (X, J) be an almost complex manifold. As mentioned above, the almost complex structure operator J splits the bundle T XC into elements of type (1, 0), eigenvectors associated to the eigenvalue i of J, and elements of type (0, 1), eigenvectors associated to the eigenvalue −i of J. The bundle T 1,0X is the complex conjugate of the bundle T 0,1X. The following theorem gives an exact description of the integrable almost complex structures. Theorem 3.14. (Newlander-Nirenberg) The almost complex structure J is integrable if and only if we have [T 0,1X, T 0,1X] ⊂ T 0,1X. Remark 3.15. By passing to the conjugate, this is equivalent to the condition that the bracket of two vector fields of type (1, 0) is of type (1, 0). This theorem is a difficult theorem in analysis, for it implies, in particular, that the manifold X which was assumed to be only differentiable actually admits the structure of a real analytic manifold.

3.16. The operators ∂ and ∂¯. 3.16.1. Definition. Let (X, J) be an almost complex manifold; the decomposition at the , , tangent bundle level T XC = T 1 0X ⊕ T 0 1X induces a dual decomposition

1,0 0,1 (3.1) ΩX,C =ΩX ⊕ ΩX . 1,0 When X is a complex manifold, the bundle ΩX of complex differential forms of type (1, 0), i.e. C-linear forms, is generated in holomorphic local coordinates z1,...,zn by the dzi, i.e. a form a of type (1, 0) can be written locally as α = i αidzi, where the ai, are ∞ ∞ C functions if α is C . Since d(dzi) = 0, it follows that dα = dαi ∧dzi. Furthermore, the decomposition 3.1 also induces the decomposition of the complexP k-forms into forms of type (p,q), for p + q = k: P k p,q (3.2) Λ ΩX,C = ΩX , p+Mq=k 20 DANIEL MATEI SNSB 2008

p,q p 1,0 q 0,1 where the bundle ΩX is equal to Λ ΩX ⊕ Λ ΩX . With this definition, formula (3.2) shows that if the almost complex structure is integrable and α is a differential form of 2,0 1,1 type (1, 0), then dα is a section of ΩX ⊕ ΩX . In fact, using the formula χ(α(η)) − η(α(χ)) = dα(χ, η)+ α([χ, η]), where α is a 1-form and χ, η are vector fields, we easily see that this property is equivalent to the integrability condition of theorem 3.14, and thus to the integrability of the almost complex structure. p,q More generally, the bundle ΩX admits as generators in holomorphic local coordinates z1,...,zn, in the differential forms

dzI ∧ dz¯J = dzi1 ∧···∧ dzip ∧ dz¯j1 ∧···∧ dz¯jq ,

where I, J are sets of ordered indices 1 ≤ i1 < · · · < ip ≤ n and 1 ≤ j1 < · · · < jq ≤ n. Note that these forms are closed, i.e. annihilated by the exterior differential operator d. A form α of type (p,q) can thus be written locally as α = I,J αI,J dzI ∧ dz¯J . It follows that P dα = dαI,J ∧ dzI ∧ dz¯J I,J X is the sum of a form of type (p,q + 1) and a form of type (p + 1,q). Definition 3.17. For a C∞ differential form a of type (p,q) on a complex manifold X, we define ∂α to be the component of type (p,q + 1) of dα. Similarly, we define ∂α¯ to be the component of type (p + 1,q) of dα. For (p,q) = (0, 0), a form of type (p,q) is a function f; ∂f¯ is then the C-antilinear part of df, and thus it vanishes if and only if f is holomorphic. By definition, we have ∂f ∂f df = dz + dz¯ , ∂z i ∂z¯ i i i i i X X and thus ∂f ∂f¯ = dz¯ . ∂z¯ i i i X As mentioned above, a differential k-form α decomposes uniquely into components αp,q of type (p,q),p + q = k. We then set ∂α¯ = ∂α¯ p,q, ∂α = ∂αp,q. p,q p,q X X The following lemmas describe the essential properties of the operators ∂ and ∂¯. Lemma 3.18. The operator ∂¯ satisfies Leibniz’ rule ∂¯(α ∧ β)= ∂α¯ ∧ β + (−1)deg αα ∧ ∂β.¯ Similarly, the operator ∂ satisfies Leibniz’ rule ∂(α ∧ β)= ∂α ∧ β + (−1)deg αα ∧ ∂β. INTRODUCTIONTOHODGETHEORY 21

Proof. The second assertion follows from the first, since by definition of the operators ∂ and ∂¯, we have the relation ∂α¯ = ∂¯α¯. As for the first relation, it suffices to prove it for α of type (p,q) and β of type (p′,q′). We then obtain it immediately in this case, by taking the component of type (p + p′,q + q′ + 1) of d(α ∧ β).  Lemma 3.19. We have the following relations between the operators ∂ and ∂¯. ∂¯2 = 0, ∂2 = 0, ∂∂¯ + ∂∂¯ = 0. Proof. This follows from the formulas d ◦ d = 0, d = ∂ + ∂¯. Indeed, these relations imply that ∂2 + ∂∂¯ + ∂∂¯ + ∂¯2 = d2 = 0. Now, if α is a form of type (p,q) then ∂2α is of type (p + 2,q), (∂∂¯ + ∂∂¯ )α is of type (p + 1,q + 1) and ∂¯2α is of type (p,q + 2). Thus, d2α = 0 implies that ∂2α = (∂∂¯ + ∂∂¯ )α = ∂¯2α = 0.  3.19.1. Local exactness. The Poincare lemma shows the local exactness of the oper- ator d (see Bott and Tu [1]): Lemma 3.20. Let α be a closed differential form of strictly positive degree on a differ- entiable manifold. Then, locally, there exists a differential form β such that α = dβ. We say that α is locally exact. Now consider a complex manifold X. Let α = dβ be a form of type (p,q) which is ∂¯-exact. Then we have ∂α¯ = 0 by lemma 3.19. The following proposition is a partial converse which is the analogue of the Poincare lemma for the operator ∂¯. Proposition 3.21. Let α be a differential form of type (p,q) with q > 0. If ∂α¯ = 0, then there locally exists on X a differential form β of type (p,q − 1) such that α = ∂β¯ . Proof. We first reduce to the case where p = 0 by the following argument. Locally, we can write in holomorphic coordinates z1,...,zn

α = αI,J dzI ∧ dz¯J , I,J X where the sets of indices I are of cardinal p and the sets of indices J are of cardinal q. Then ∂α¯ = ∂α¯ I,J ∧ dzI ∧ dz¯J I,J X by lemma 3.18. It follows that if ∂α¯ = 0, for every I of cardinal p the form αI of type (0,q) defined by αI = dz¯J J X is ∂¯-closed. If the proposition is proved for forms of type (0,q), then locally we have αI = ∂β¯ I , and

p α = (−1) ∂¯ dzI ∧ βI . I ! X It remains to show the proposition for forms of type (0,q). Such a form can be written α = J αJ dz¯J . We do the proof by induction on the largest integer k such that there P 22 DANIEL MATEI SNSB 2008 exists J with k ∈ J and αJ 6= 0. Necessarily k ≥ q. If k = q, we have a = fdz¯1 ∧···∧ dz¯q. The condition ∂α¯ = 0 is then equivalent to the fact that the function f is holomorphic in the variables zl,l>q. We then apply Cauchy’s theorem : note that its proof also gives the following. 

Proposition 3.22. Let f(z1,...,zn) be a differential function which is holomorphic in the variables zl,l>q. Then there locally exists a differential function g, holomorphic in the variables z ,l>q, such that ∂g = f. l ∂z¯q 3.22.1. Dolbeault complex of a holomorphic bundle. Let E be a holomorphic vector bundle of rank over a complex manifold X. Let E0,q(E) denote the space of C∞ sections 0,q Cn of the bundle ΩX ⊗C E. In a holomorphic trivialisation of E, φU : E|U = U × , such ∞ a section can be written (α1,...,αk), where the αi, are C forms of type (0,q) on U. ¯ ¯ ¯ 0,q+1 We then set ∂U α = (∂α1,..., ∂αk); it is a section of ΩU ⊗C E. We will show that this local definition in fact gives a form ∂α¯ ∈E0,q+1(E). Ck Lemma 3.23. Let V be an open subset of X and φV : E|V = V × a holomorphic trivialisation of E over V . Then for α ∈E0,q(E), we have

∂¯U α |U∩V = ∂¯V α |U∩V .

Proof. Let MUV be the transition matrix, with holomorphic coefficients, which enables us to pass from the trivialisation φU to the trivialisation φV . Then, by definition, if αU is a section of E over U, αU = (α1,U ,...,αk,U ) in the trivialisation φU , and αV is a section of E over V , αV = (α1,V ,...,αk,V )in the trivialisation φV , the sections αU and αV coincide on U ∩ V if and only if T T (α1,V ,...,αk,V ) = MUV (α1,U ,...,αk,U ) .

We can of course replace the functions αi by differential forms. The form α can be written (α1,U ,...,αk,U ) in the trivialisation φU and (α1,V ,...,αk,V ) in the trivialisation φV , and we have, as above, T T (α1,V ,...,αk,V ) = MUV (α1,U ,...,αk,U ) . To see ∂¯U α |U∩V = ∂¯V α |U∩V . by the above and the definition of ∂¯U , ∂¯V , it suffices to show that T T (∂α¯ 1,V ,..., ∂α¯ k,V ) = MUV (∂α¯ 1,U ,..., ∂α¯ k,U ) . But this follows immediately from the Leibniz formula lemma 3.18 and the fact that the matrix MUV has holomorphic coefficients.  Lemma 3.23 enables us to define an operator 0,q 0,q+1 ∂¯E : E (E) →E (E)

by the condition ∂¯Eα |U = ∂¯U α |U . Note that the meaning of this operator on the space E0,0(E) of C∞ sections of E is the following. INTRODUCTIONTOHODGETHEORY 23

Lemma 3.24. The 0,0 0,1 Ker ∂¯E : E (E) →E (E) contains exactly the holomorphic sections of E.  Proof. This is clear, since the holomorphic sections are those which are given by n-tuples of holomorphic functions in local holomorphic trivialisations. But by definition, ∂¯E acts like the operator ∂¯ on these n-tuples, and we know that the functions annihilated by ∂¯ are exactly the holomorphic functions.  Naturally, this operator satisfies the same local properties as the operator ∂¯ on the forms.

Lemma 3.25. The operator ∂¯E satisfies Leibniz’ rule q ∂¯E(α ∧ β)= ∂¯Eα ∧ β + (−1) α ∧ ∂¯Eβ. Here, α is a differential form of type (0,q), and β is a differential form of type (0,q′) with coefficients in E, so that α ∧ βis naturally a differential form of type (0,q + q′) with coefficients in E. ¯ ¯2 Clearly, the operator ∂E also satisfies the property ∂E = 0. Finally, the local exactness of ∂¯E follows from that of the operator ∂¯. Proposition 3.26. Let α be a form of type (0,q) with coefficients in E, and q > 0. If ∂¯Eα = 0, then locally on X there exists a form β of type (0,q − 1) with coefficients in E such that α = ∂¯Eβ.

Recall that a complex (of vector spaces for example) is a family of vector spaces Vi together with morphisms di : Vi → Vi+1 satisfying di+1 ◦ di = 0. The standard example i is the de Rham complex of a differentiable manifold X, where Vi = E (X) is the space of differential forms of degree i, and di = d. Definition 3.27. The complex ¯ ¯ 0,∗ 0,q−1 ∂E 0,q ∂E 0,q+1 E (E), ∂¯E = ···→E (E) →E (E) →E (E) → . . .   is called the Dolbeault complex of E. 3.28. Examples of complex manifolds. 3.28.1. Riemann surfaces. Let us consider 2-dimensional differentiable manifolds. If we restrict ourselves to the compact oriented case, these manifolds are classified by their genus g: such a surface is diffeomorphic to the g-holed torus. Furthermore, we can always put complex structures on such surfaces X. Indeed, we first note that the Newlander- Nirenberg integrability condition is automatically satisfied by an almost complex structure on X, by the fact that the rank of the complex vector bundle T 0,1X is equal to 1, and the bracket of vector fields is alternating. Thus, every almost complex structure is induced by a complex structure. Moreover, the existence of almost complex structures follows from the existence of Riemannian metrics on X: An almost complex structure on an oriented 24 DANIEL MATEI SNSB 2008 surface X is equivalent to a conformal structure on X, i.e. to a Riemannian metric on X defined up to multiplication by a positive function. Compact Riemann surfaces are called curves in algebraic geometry. Indeed, they are 1-dimensional complex manifolds (varieties). 3.28.2. Complex projective space. The complex projective space CPn is the set of complex lines of Cn+1, or equivalently, the quotient of Cn+1 \0 by the equivalence relation identifying collinear vectors on C. The topology is the quotient topology. The complex n+1 structure is obtained as follows. For each i, consider the open subset U˜i of C \ 0 n consisting of the points z such that zi 6= 0. Let Ui be the image of U˜i in CP . Each point z ∈ Ui admits a unique lifting z to U˜i which satisfies the condition zi = 1. Thus, Ui is naturally homeomorphic to Cn, which provides the holomorphic charts for CPn, which is covered by the Ui. It remains simply to check that the change of chart morphisms are holomorphic. But Ui ∩ Uj can obviously be identified with the classes of non-zero n+l vectors z ∈ C such that zi 6= 0 and zj 6= 0. Given such a vector, the image of the representative of its class in the chart U = Cn is given by z1 ,..., 1,..., zn+1 , where i zi zi the 1 is in the ith place, while the image of the representative of its class in the chart U = Cn is given by z1 ,..., 1,..., zn+1 , where the 1 is in the jth place. The transition j zj zj morphism is thus given, up to the order of the coordinates, by 1 ζ ζ (3.3) (ζ ,...,ζ ) → , 1 ,..., n 1 n ζ ζ z  j j j  n n on C \ {ζj = 0}. As (3.3) is clearly holomorphic, we have equipped CP with a complex structure. 3.28.3. Complex tori. Let Γ be a in Cn, i.e. a free additive subgroup generated by a basis of Cn over R. The group Γ acts by translation on Cn, and the action is proper and fixed-point-free. The quotient T = Cn/Γ is compact. In fact, there exists a R-linear automorphism of Cn = R2n sending Γ to Z2n so that this quotient is naturally homeomorphic to (R/Z)n = (S1)n. Clearly, T admits a natural differentiable structure for which the quotient map is a local diffeomorphism. We then put an almost complex structure onto T by taking the holomorphic charts to be the local inverses of the quotient map. As these local inverses are defined up to translation by an element of Γ, the change of chart morphisms are given by these translations, which are obviously holomorphic. Thus, T is equipped with a holomorphic structure. INTRODUCTIONTOHODGETHEORY 25

References

[1] R. Bott, L. Tu, Differential forms in . [2] P. Griffiths, J. Harris, Principles of algebraic geometry. [3] A. Hatcher, Algebraic Topology, available at http://www.math.cornell.edu/~hatcher/ [4] C. Voisin, Hodge theory and complex algebraic geometry. [5] F. Warner, Foundations of differentiable manifolds and Lie groups.

Institute of Simion Stoilow of the Romanian Academy E-mail address: [email protected]